Advanced Sudoku Strategy

Now that you have mastered the basic strategies you can tackle almost any Sudoku puzzle. With the full range of tactics even puzzles categorised as 'fiendish' can be solved. And yet one of the delights of Sudoku is that there is another level of advanced strategies to call upon that distinguish the Sudoku Masters from the rest. It is amazing how such an apparently simple puzzle can lead to such complexity.

Colorful Puzzles

When describing the X-Wing and Swordfish the strategy moved into a more complex area. The tactics involved looking beyond one or two groups of squares to several interlocking groups. Both these strategies rely on pairs of squares. This is a powerful technique that investigates both allocation options and finds squares that are common to both alternates. This pair rule only applies for squares where one number can only go in one of two squares in a single group (but this may be a row; column or region), if there are three or more it can't be used.

Here is an example of an X-Wing. There are two rows where there are only two squares which can take a 4 (rows A and H_ These match up by columns to form a box. There are only two alternative allocations for 4 in Ac and Hh or Ah and Hc. This is conveniently illustrated by using alternate blue and orange coloring for the squares. Either the blue squares take a 4 or the orange squares take a 4. This is useful because it's now unquestionable that a 4 occurs in both column c and h so all the squares that also looked as though they could have taken a 4 in these two columns can be discounted (shown in pink). Using alternate colors helps to show what is going on. Sudoku Dragon supports six different colors for marking squares.

 To download this puzzle  click here

So now let's turn to the Swordfish to see if coloring helps there too. In this Swordfish there are three rows (A, B and H) all with a pair possible squares for a 6. As the squares are all located in just three columns b; g and i they form a 'Swordfish'. The 6s must either go on the blue squares or the orange squares they can not be allocated any other way. Because the three columns b; g and i all contain both a blue and orange square then a 6 can not go in any other square in these columns. So the pink square Gi can not actually take a 6 even though it looks as though it could do. Once again coloring the pairs has made it easy to see the options.

Alternate Pair Exclusion

The X-Wing and Swordfish are just examples of a more general rule. This is always satisfying for a mathematician, by looking at simple examples it is now possible to extend the same reasoning to general cases.

If you identify pairs of squares in the same row; column or region for the same number you can start coloring them. If you find another linked pair sharing any square with a colored pair then you can continue using the same alternate coloring scheme for that pair too. These are termed 'alternate pairs' as the true allocation must be in one of the alternative squares in the pair, you will sometimes see these called 'conjugate pairs', this term comes from mathematics and is not as descriptive.

When you have finished coloring the linked pairs you can then look at the pattern and make use of the implications. If you find a group (row; column or region) that has both colors in any of its squares then any other possibility for that number can be safely excluded from the group. This general rule covers not only the X-Wing and Swordfish types but a whole range of other patterns of possibilities too.

Here is a grid with the possibilities for 6 colored. There are pairs of possibilities for 6 in row B and C as well as region Ag (it is often a region pair that is helpful). There is also a pair in column b that does not interlink with another pair and so does not help. The pairs have been colored and this is useful because column a has both an orange and a blue square in it so the 'possible' 6 in square Ha (pink) can be excluded. For more on Sudoku Dragon's support see our square coloring page.

Sudoku Dragon highlights alternate pairs and uses the alternate pair strategy to solve puzzles.

 To download this puzzle  click here

You will often find that using alternate pair coloring does not actually help solve any more squares so it is a technique to hold in reserve for very tricky puzzles because it takes a while to work through. Pairs are most frequently found whether you have a dozen or so possible squares left for a number, they are not common at the start or end of Sudoku puzzle solving.

Alternate Pair Deduction

Just when you thought strategies were becoming too mind blowing there is always a new and tantalizing twist. It is another good reason to get out your crayons and start coloring Sudoku squares. It is slightly different to the previous section on 'alternate pair exclusion', the pairs are identified and colored in just the same way but in this case the implication is different and often much more useful. If after coloring you end up with any region with two squares of the same color then something is distinctly odd. If the coloring produces two or more squares of the same color then this color assignment is not possible and the other color is the correct one and all those squares can all be set as the only possible option for the number.

Starting with orange for the square Fd and blue for its alternate row pair Fe, the pairs in column e come into play and orange goes into Ae. From Fd there is a pair in column d so blue for Hd, then following the pair in row I orange goes in Ia, now there is a pair in region Ga which means Gb must be blue, finally column b is a pair so Ab is orange too. So finally we end up with two orange squares in row A, which can not be right. If the 7s went in the orange squares there would be two 7s in row A. So the 7s can not go in the orange squares they must go in the blue squares Fe; Gb and Id.

 To download this puzzle  click here

For an extensive run-through example of this strategy  click here

Multicolored Alternate Pairs

Once you have mastered Alternate Pairs it is worth introducing a further complication that may be useful. When you color a grid you will quite often find there are two independent chains of interlinked pairs. You can use two more colors to mark up the other set of pairs and you have more than just a very colorful grid! The way the different networks of linked colored squares can restrict and force square allocations as in the case of a single network of pairs.

Here's a puzzle with all the squares that can take a 6 have been marked using Sudoku Dragon. There are some useful looking pairs. In region Dg the orange square Eh and blue square Fg which links on Row F to orange pair Ff, and Eh links through column h to blue square Gh. Square Fg links through column g to orange Hg. That's it for that set but there is another pair in column c. Square Gc has been colored red and Ic green. Now the crucial fact is that the two sets are related via row G (the possibility in Gd forbids the two sets being fully interlinked). If a 6 were to go in the blue squares it could not go in the red square it would have to go in the green square Ic. Alternatively if a 6 was correct for the orange squares then this does not intersect with the red-green pair directly so it can't be deduced which one is correct. However, putting the alternatives together where orange and green squares intersect its now certain that a 6 can't go, and in this example it is square If marked in pink. Either a 6 is in orange square Ff or it is in green square Ic so both possible cases forbid a 6 in the pink square which is common between this row and column.

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Further twists

What was achieved with pairs could equally be achieved with triplets or quadruplets. All that is needed is to work through the logic and see if all the three or four alternatives all combine to have a common implication. However these are too rare and too hard to identify to be of practical use.

Hook or X-Y Wing

When I first saw the term X-Y Wing I thought someone had just mistyped X-Wing or that it was yet another variant of it. However, the X-Y Wing is another tricky advanced strategy separate from pairs but once again using the 'either-or' logic that flows from using connected groups.

To avoid confusion we'll use the term Hook for this technique. Under rather special circumstances, the Hook knocks out other possibilities. An hook (or X-Y Wing) requires you to find three squares. The squares must have two possibilities each in three different numbers. The three squares form a chain of pairs of possibilities. An example of such a chain is [2; 5] [5;7] and [7;2] expressed this way you can see the chain of possibilities is [2 -> 5 -> 7 -> 2]. Where these squares are located is also important, two must be in the same row or column, they form the stem and the other must be in the same region as one of the other two squares.

It is all too easy to slip up on correct identification of where a Hook is located. How is this obscure relationship useful? Well the way these squares is arranged restricts possibilities elsewhere in the grid. If in our example the stem contains [2,5] and [5,7] then 5 is the stem number and the [2,7] the branch or hook. If 5 is the correct choice for the [2,5] square then that means the [5,7] square must be 7. The only alternative for [2,5] square is 2, a little logical magic now forces the [2,7] square to be 7 and therefore the [5,7] square must be 5. These are the only two choices and if there are any squares where a 7 is not possible for both these two alternatives then we can safely exclude 7 as a possibility for them.

Here are some pictures of a real puzzle that should help. It is a puzzle at a stage where the easy squares have run out.

The light green squares are of interest, they are Bd that can be only [2,7]; Ce that can take [2,5] (the 9 is not possible because of the naked twin [4,9] in row C) and Ch that can take [5,7] ) (the 9 is excluded for the same reason). So we have our chain [2,7] [7,5] [5,2] of three squares. The last two form the stem of the hook and 7 is the hook number. Following our logic if the 2 went in Ce this means a 7 must go in Bd and a 5 in Ch. So this gives us the following segment of the grid.

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The green squares are the ones we have tentatively allocated and the blue squares indicate where another 7 can not be put in this scenario.

The only other option for Ce was for it to be a 5 rather than a 2, and this forces us to put a 7 in Ch. So now we have the alternative scenario with Ch and Ce highlighted in green and for this alternative the blue squares show where a 7 can no longer be put.

Well, we are in luck as there is one common square in blue in both alternatives, square Bh (dark blue). It can not be a 7 for either of the two possibilities for Ce and so 7 can be safely excluded from its possibilities. For this puzzle this is crucial as that leaves only one choice for Bh which is 9, the hook strategy enables us to immediately solve a square and it turns out to be the last tricky one to solve.

The Hook is a general technique, the term X-Y Wing name comes from its mathematical formulation as three squares containing [x, y]; [y, z] and [x, z]. (In our case x=5; y=2; z=7). It tells us that the shared squares where the two alternative allocations for z intersect can not possibly contain a z.

For an extensive run-through example of this strategy  click here

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Have you mastered all the strategies for solving Sudoku? Our strategy page gives an introduction to all the main tactics for solving a Sudoku puzzle. We have separate pages on Trial and error and Advanced Strategies.. Read more...

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